## Properties, Worlds and Propositions

March 28, 2008

Two different properties can have the same extension – examples should be familiar, for example the property has a kidney is different from the property has a heart, yet they have the same extension. The fact that they are distinct is usually established by considering their modal properties. For example, having a heart could have had an instant that didn’t instantiate having a kidney and vice versa. This is certainly a sufficient condition for properties to be distinct, but not a necessary one, again examples should be familiar: being triangular and being trilateral are standard examples, supposedly different, yet necessarily coextensive.

I think this second point can be made more clearly when we consider properties for which it is quite hard to make sense of the property ‘being had in a world’. For example exists in more than two worlds and exists in more than three worlds. Presumably I have both these properties, but I’m not sure if I have them in any world in particular! Either way, they are two distinct properties. Now consider properties of worlds, for example the property a world has iff donkeys talk in that world, and the one a world has iff pigs fly in that world. Here again, we can’t talk about about these properties coming apart in different worlds because nothing has these properties in a world. But clearly these properties are different – in this case they have different extensions because there are worlds in which donkeys talk, but pigs are earthbound.

Where am I going with this? Well, once we have the following two assumptions we can solve a difficult problem in the philosophy of language. The assumptions are:

• You can have two distinct yet coextensive properties.
• Properties aren’t in any sense parasitic on worlds (so we can make sense of worlds having properties.)

The puzzle is to do with propositional attitudes. Often propositions are treated as sets of worlds. But if this is the case then we cannot explain the semantic difference between “Hesperus is Phosphorus” and “Hesperus is Hesperus”, e.g. when they appear in belief reports. Why? Because the set of worlds they each represent are coextensive, and whenever you have two coextensive sets they are identical. Thus they express the same proposition.

My idea was, instead identifying propositions with sets of worlds, to identify propositions with properties of worlds. In this case the property a world has iff Hesperus is Phosphorus in that world can be different from the property a world has iff Hesperus is Hesperus in that world, even though they are coextensive! Do we even need to postulate necessarily coextensive distinct properties to make this work? I think not – it doesn’t make sense to talk about world properties being necessarily coextensive – these two properties aren’t necessarily coextensive because their extensions aren’t world dependent.

Maybe I could have made the point better by talking in terms of world indexed properties. “Monadic” properties like being bent are actually relational, with an argument place for a world. To have the property of being bent at a world w is just to have the monadic property $\lambda x.bent(x, w)$ but if you consider the relation $\lambda x\lambda y.bent(x, y)$ it doesn’t make too much sense to say it is had by a pair of objects at some world, as opposed to any other. For a world indexed property to be had at a world by o is just for o to hold the corresponding relation to w. If the property is genuinely monadic, or has no world argument place then the talk of properties holding at a world falls down on this view.